The results of a national survey showed that on average, adults sleep 6.9 hours

Question

The results of a national survey showed that on average, adults sleep 6.9 hours per night. Suppose that the standard deviation is 1 hours. Round your answers to the nearest whole number.

Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 4.9 and 8.9 hours.

22

At least % Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 hours.

At least % Assume that the number of hours of sleep follows a bell-shaped distribution.

Use the empirical rule to calculate the percentage of individuals who sleep between 4.9 and 8.9 hours per day.

% How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)?

Explanation / Answer

Answer in detail:

1.

4.9 and 8.9 are 2 deviations above and below mean of 6.9, since the standard deviation is 1
So, According to Chebyshev' therorm, area under k deviations from mean is given by : 1-1/k^2
So, 1-1/2^2 = 75% is the total %age of individuals who sleep between 4.9 and 8.9

2.

3.9 and 9.9 is 3 deviations from mean, total %age of individuals who sleep between 3.9 and 9.9 = 1-1/3^2 = 8/9 or 88.89%

3. Now when we are asked to assume bell curve distribution, we can use the fact that 68-95-99.7% of area is under 1-2-3 deivations from mean.

So, 4.9 and 8.9 is 2 deviations, ie. 95% of area is covered between 4.9 and 8.9

a) Chebeshev' gave us an area of 88.89%, whereas a normal distribution is more bell type, covering more around the mean. It covers 95% instead of Chebeshev' 88.89%

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